On the classification of quadratic forms over an integral domain of a global function field

Let C be a smooth projective curve defined over the finite field Fq (q is odd) and let K=Fq(C) be its function field. Any finite set S of closed points of C gives rise to an integral domain OS:=Fq[C−S] in K. We show that given an OS-regular quadratic space (V,q) of rank n≥3, the set of genera in the proper classification of quadratic OS-spaces isomorphic to (V,q) in the flat or étale topology, is in 1:1 correspondence with Br2(OS), thus there are 2|S|−1 genera. If (V,q) is isotropic, then Pic (OS)/2 classifies the forms in the genus of (V,q). For n≥5, this is true for all genera, hence the full classification is via the abelian group Hét2(OS,μ_2).